# Cut-and-join structure and integrability for spin Hurwitz numbers

## TL;DR

This paper explores the algebraic structure and integrability properties of spin Hurwitz numbers, revealing their connection to Q Schur functions, Sergeev characters, and BKP hierarchy, with implications for matrix models.

## Contribution

It introduces a cut-and-join structure for spin Hurwitz numbers and demonstrates their relation to integrable BKP hierarchy and Sergeev characters.

## Key findings

- Spin Hurwitz numbers are eigenfunctions of cut-and-join operators.
- The generating function is a BKP tau-function.
- Relations to matrix models are discussed.

## Abstract

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators W_\Delta with \Delta\in OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a \tau-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.